Prove that: \[{{\left( \sin \theta +\cos ec\theta \right)}^{2}}+{{\left( \cos \theta +\sec \theta \right)}^{2}}=7+{{\tan }^{2}}\theta +{{\cot }^{2}}\theta \].

Hint: In this question we will use some basic trigonometric formulas and applications of formulas to solve this question. Before solving this question, we have to know that \[{{\sin }^{2}}\theta +{{\cos }^{2}}\theta =1\] , \[{{\sec }^{2}}\theta -{{\tan }^{2}}\theta =1\],\[\cos e{{c}^{2}}\theta -{{\cot }^{2}}\theta =1\] also we will use one basic algebraic formula \[{{\left( a+b \right)}^{2}}={{a}^{2}}+{{b}^{2}}+2ab\].

Complete step-by-step solution:
From the question it is clear that we have to prove that: \[{{\left( \sin \theta +\cos ec\theta \right)}^{2}}+{{\left( \cos \theta +\sec \theta \right)}^{2}}=7+{{\tan }^{2}}\theta +{{\cot }^{2}}\theta \].
So, now let us try to prove by taking LHS part i.e \[{{\left( \sin \theta +\cos ec\theta \right)}^{2}}+{{\left( \cos \theta +\sec \theta \right)}^{2}}\]
\[\Rightarrow {{\left( \sin \theta +\cos ec\theta \right)}^{2}}+{{\left( \cos \theta +\sec \theta \right)}^{2}}\]………………(1)
Now let us try to simplify the equation by expanding the square terms.
Looking carefully, we can see that it is in the form of \[{{\left( a+b \right)}^{2}}\].
Now using basic algebraic formula \[{{\left( a+b \right)}^{2}}={{a}^{2}}+{{b}^{2}}+2ab\] we will expand the square terms.
So now, after expanding we can write
\[\Rightarrow {{\left( \sin \theta +\cos ec\theta \right)}^{2}}={{\sin }^{2}}\theta +\cos e{{c}^{2}}\theta +2\sin \theta \cos ec\theta \]………………..(2)
\[\Rightarrow {{\left( \cos \theta +\sec \theta \right)}^{2}}={{\cos }^{2}}\theta +{{\sec }^{2}}\theta +2\cos \theta \sec \theta \]…………………...(3)
Now put equations (2) and (3) in equation (1).
So, equation (1) becomes
\[\Rightarrow {{\left( \sin \theta +\cos ec\theta \right)}^{2}}+{{\left( \cos \theta +\sec \theta \right)}^{2}}={{\sin }^{2}}\theta +\cos e{{c}^{2}}\theta +2\sin \theta \cos ec\theta +{{\cos }^{2}}\theta +{{\sec }^{2}}\theta +2\cos \theta \sec \theta \]
Now we have reduced the equation into the simplest form.
Let us rearrange the terms for the simple calculation
\[\Rightarrow {{\left( \sin \theta +\cos ec\theta \right)}^{2}}+{{\left( \cos \theta +\sec \theta \right)}^{2}}={{\sin }^{2}}\theta +{{\cos }^{2}}\theta +\cos e{{c}^{2}}\theta +{{\sec }^{2}}\theta +2\sin \theta \cos ec\theta +2\cos \theta \sec \theta \]
We already know that \[{{\sin }^{2}}\theta +{{\cos }^{2}}\theta =1\].
So,
\[\Rightarrow {{\left( \sin \theta +\cos ec\theta \right)}^{2}}+{{\left( \cos \theta +\sec \theta \right)}^{2}}=1+\cos e{{c}^{2}}\theta +{{\sec }^{2}}\theta +2\sin \theta \cos ec\theta +2\cos \theta \sec \theta \].
From the basic trigonometric conversions, we can write \[\sin \theta =\dfrac{1}{\cos ec\theta }\] and \[\cos \theta =\dfrac{1}{\sec \theta }\].
Now the equation becomes,
\[\Rightarrow {{\left( \sin \theta +\cos ec\theta \right)}^{2}}+{{\left( \cos \theta +\sec \theta \right)}^{2}}=1+\cos e{{c}^{2}}\theta +{{\sec }^{2}}\theta +2\times \dfrac{1}{\cos ec\theta }\times \cos ec\theta +2\times \dfrac{1}{\sec \theta }\times \sec \theta \]
On simplification, we get
\[\Rightarrow {{\left( \sin \theta +\cos ec\theta \right)}^{2}}+{{\left( \cos \theta +\sec \theta \right)}^{2}}=1+\cos e{{c}^{2}}\theta +{{\sec }^{2}}\theta +2+2\]
\[\Rightarrow {{\left( \sin \theta +\cos ec\theta \right)}^{2}}+{{\left( \cos \theta +\sec \theta \right)}^{2}}=5+\cos e{{c}^{2}}\theta +{{\sec }^{2}}\theta \]………………(4)
Now from the basic trigonometric formulas we can write \[{{\sec }^{2}}\theta ={{\tan }^{2}}\theta +1\] and \[\cos e{{c}^{2}}\theta ={{\cot }^{2}}\theta +1\]. Put these values in equation (4).
\[\Rightarrow {{\left( \sin \theta +\cos ec\theta \right)}^{2}}+{{\left( \cos \theta +\sec \theta \right)}^{2}}=5+{{\cot }^{2}}\theta +1+{{\tan }^{2}}\theta +1\]
After simplification we get
\[\Rightarrow {{\left( \sin \theta +\cos ec\theta \right)}^{2}}+{{\left( \cos \theta +\sec \theta \right)}^{2}}=7+{{\cot }^{2}}\theta +{{\tan }^{2}}\theta \].
Now we got the simplified answer.
Hence, we have proved that \[{{\left( \sin \theta +\cos ec\theta \right)}^{2}}+{{\left( \cos \theta +\sec \theta \right)}^{2}}=7+{{\cot }^{2}}\theta +{{\tan }^{2}}\theta \].

Note: students should be able to use proper formulas and applications of this formulas while solving this type of questions. In case of any use of wrong formulas may lead to do this type of questions wrong. students should avoid calculation mistakes while solving \[{{\left( \sin \theta +\cos ec\theta \right)}^{2}}+{{\left( \cos \theta +\sec \theta \right)}^{2}}=7+{{\cot }^{2}}\theta +{{\tan }^{2}}\theta \].