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'''a tanθ+b secθ=c''' সমীকরণটিকে আমরা এভাবে লিখতে পারি:

'''b secθ = c - a tanθ'''

উভয় পক্ষকে বর্গ করে পাই:

'''b² sec²θ = (c - a tanθ)²'''

আমরা জানি, '''sec²θ = 1 + tan²θ''', তাই:

'''b²(1 + tan²θ) = c² - 2ca tanθ + a² tan²θ'''

'''b² + b² tan²θ = c² - 2ca tanθ + a² tan²θ'''

সব পদ একপাশে এনে '''tanθ''' এর সাপেক্ষে একটি দ্বিঘাত সমীকরণ গঠন করি:

'''a² tan²θ - b² tan²θ - 2ca tanθ + c² - b² = 0'''

'''(a² - b²) tan²θ - 2ca tanθ + (c² - b²) = 0'''

এই দ্বিঘাত সমীকরণের মূলদ্বয় '''tanα''' এবং '''tanβ'''।

মূলদ্বয়ের যোগফল:

'''tanα + tanβ = -(-2ca) / (a² - b²) = 2ca / (a² - b²)'''

মূলদ্বয়ের গুণফল:

'''tanα tanβ = (c² - b²) / (a² - b²)'''

এখন, আমরা '''tan(α + β)''' এর সূত্র ব্যবহার করব:

'''tan(α + β) = (tanα + tanβ) / (1 - tanα tanβ)'''

মানগুলো বসিয়ে পাই:

'''tan(α + β) = [2ca / (a² - b²)] / [1 - (c² - b²) / (a² - b²)]'''

'''tan(α + β) = [2ca / (a² - b²)] / [(a² - b² - (c² - b²)) / (a² - b²)]'''

'''tan(α + β) = [2ca / (a² - b²)] / [(a² - b² - c² + b²) / (a² - b²)]'''

'''tan(α + β) = [2ca / (a² - b²)] / [(a² - c²) / (a² - b²)]'''

'''ta

(a tanθ+b secθ=c ) সমীকরণের মূলদ্বয় ( alpha ), ( beta ) হলে, প্রমাণ করে যে, ( tan left ( alpha + beta right ) = 2 c a a^ 2 - c^ 2 ).